Question

A certain club has 10 members, including Harry. One of the 10 members is to be chosen at random to be the president, one of the remaining 9 members is to be chosen at random to be the secretary, and one of the remaining 8 members is to be chosen at random to be the treasurer. What is the probability that Harry will be either the member chosen to be the secretary or the member chosen to be the treasurer?

• A. \(\frac{1}{720}\)
• B. \(\frac{1}{80}\)
• C. \(\frac{1}{10}\)
• D. \(\frac{1}{9}\)
• E. \(\frac{1}{5}\)

Hint:

For problems involving selecting distinct roles, remember that any individual has an equal chance of landing in any specific role. The probability of a person being chosen for any one of N roles is 1/Total People. If the question asks for the probability of them being in one of K possible roles, the answer is often K/Total People.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem asks for the probability of an event that can occur in two mutually exclusive ways (Harry can be secretary, or Harry can be treasurer, but not both). We can calculate the probability of each case and then add them together.
Step 2: Key Formula or Approach:
The probability of "Event A OR Event B" where A and B are mutually exclusive is \(P(A \text{ or } B) = P(A) + P(B)\).
We will calculate the probability of Harry becoming secretary and the probability of Harry becoming treasurer separately.
Step 3: Detailed Explanation:
Method 1: Direct Logic
There are 3 distinct positions (President, Secretary, Treasurer). Each of the 10 members has an equal chance of being selected for any specific position.
The probability that Harry is chosen for the President position is \(\frac{1}{10}\).
The probability that Harry is chosen for the Secretary position is also \(\frac{1}{10}\).
The probability that Harry is chosen for the Treasurer position is also \(\frac{1}{10}\).
The question asks for the probability that Harry is chosen as secretary OR treasurer. Since these are mutually exclusive events, we add their probabilities: \[ P(\text{Harry is Secretary or Treasurer}) = P(\text{Harry is Secretary}) + P(\text{Harry is Treasurer}) \] \[ P = \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = \frac{1}{5} \] Method 2: Step-by-Step Probability Calculation
Case 1: Harry is chosen as Secretary.
- The president must be someone other than Harry. The probability of this is \(\frac{9}{10}\).
- Given that someone else is president, there are 9 members left. The probability that Harry is chosen as secretary is \(\frac{1}{9}\).
- \(P(\text{Harry is Secretary}) = P(\text{Not Harry as President}) \times P(\text{Harry as Secretary}) = \frac{9}{10} \times \frac{1}{9} = \frac{1}{10}\).
Case 2: Harry is chosen as Treasurer.
- The president is not Harry (probability \(\frac{9}{10}\)).
- The secretary is not Harry (given the president was not Harry). There are 9 members left, 8 of whom are not Harry. Probability is \(\frac{8}{9}\).
- The treasurer is Harry. There are 8 members left. Probability is \(\frac{1}{8}\).
- \(P(\text{Harry is Treasurer}) = \frac{9}{10} \times \frac{8}{9} \times \frac{1}{8} = \frac{1}{10}\).
Total Probability:
\[ P(\text{Total}) = P(\text{Case 1}) + P(\text{Case 2}) = \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = \frac{1}{5} \] Step 4: Final Answer:
The probability that Harry will be either the secretary or the treasurer is \(\frac{1}{5}\).